Diffusion Onsager Coefficients Lij for the NaCl + Na2SO4 + H2O

636

J. Chem. Eng. Data 2009, 54, 636–651

Diffusion Onsager Coefficients Lij for the NaCl + Na2SO4 + H2O System at 298.15 K† Joseph A. Rard* 4363 Claremont Way, Livermore, California 94550

John G. Albright‡ Department of Chemistry, Texas Christian University, Ft. Worth, Texas 76129

Donald G. Miller§ 2862 Waverly Way, Livermore, California 94551

The isothermal diffusion Onsager coefficients Lij have been calculated and the Onsager reciprocal relation (ORR) has been tested for 23 compositions of the NaCl + Na2SO4 + H2O system at 298.15 K. The calculations are based on our previously reported extensive set of Fick’s law based diffusion coefficients measured using the high-quality Gosting diffusiometer and volumetric measurements made with a vibratingtube densimeter or (at a few compositions) with pycnometry. These calculations also require the four chemicalpotential concentration derivatives, which were obtained by reanalysis of available isopiestic data using a hybrid thermodynamic model. This model uses an extended form of Pitzer’s ion-interaction model to represent the single-salt thermodynamic activities but uses the mixing terms from Scatchard’s neutral-electrolyte model. Because of the ORR, the two cross-term diffusion Onsager coefficients should be equal, i.e., L12 ) L21, on either the solvent-fixed or volume-fixed reference frames. The ORR was actually tested using the experimental volume-fixed diffusion coefficients, with an equation that reduces the propagation-of-errors calculations by removing unnecessary terms. The ORR is found to be satisfied for 22 of the 23 compositions when estimated errors (usually 5 to 10 % for the chemical-potential derivatives, depending on the composition, 4 times the standard errors for diffusion coefficients or scatter observed from cross-plots, whichever is larger) are assigned for the input quantities.

Introduction Diffusion of a solute i in a binary solution without bulk solution flow can be described by Fick’s second law (formulated in 1855)1 for non-steady-state diffusion

(Ji)R ) -(Di)R ∇ Ci

(1)

where (Ji)R is the diffusional flow, (Di)R is the binary-solution diffusion coefficient, Ci is the molar concentration of this solute, and the subscript R refers to an arbitrary reference frame. Onsager and Fuoss,2 for the solvent-fixed reference frame, and Onsager3 proposed that Fick’s second law be extended to an n-component system in the form of n-1

(Ji)R ) -

∑ (Dij)R ∇ Cj

(2)

j)1

where the flow of component i not only depends on its own concentration gradient but also is coupled to the concentration gradients of the other solutes. The summation is not over all n components because the diffusion coefficient of the component chosen to be the solvent is not independent of those of the †

Part of the special issue “Robin H. Stokes Festschrift”. * To whom correspondence should be addressed. E-mail: solution_ [email protected]. ‡ E-mail: [email protected]. § E-mail: [email protected].

solutes, which is a consequence of the choice of a reference frame.4,5 The (Dii)R are called main-term diffusion coefficients, the (Dij)R (i * j) are called cross-term diffusion coefficients, and their numerical values depend on the reference frame R. Reference frames are important in describing diffusion, and equations are available for transforming quantities from one frame into those of another.4,5 We report results in both the volume-fixed and solvent-fixed frames. Most experimental measurements are done in the volumefixed reference frame, which is the case when the concentration differences are small and the diffusion cell is closed at one end. For this reference frame, the diffusion coefficient matrix of eq 2 has real, positive eigenvalues and its determinant is positive. These are the criteria required for stable diffusion boundaries (see original references cited in ref 4). When necessary, the volume-fixed flows, diffusion coefficients, and diffusion Onsager coefficients will have the subscript V. The solvent-fixed reference frame gives a simpler representation in the linear irreversible thermodynamics description of diffusion and is the best for the theoretical description of electrolyte systems. When necessary, the solvent-fixed flows, diffusion coefficients, and diffusion Onsager coefficients will have the subscript 0. In the following discussion, we will emphasize ternary solutions of two electrolytes with a common ion in a neutral solvent (in this case H2O) and will denote the two solutes as

10.1021/je800725k CCC: $40.75  2009 American Chemical Society Published on Web 12/29/2008

Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 637

components 1 and 2 and the solvent as component 0. (Note that if the two electrolytes do not share a common ion, then they form a quaternary system for diffusion and therefore have nine diffusion coefficients.) The above equations were written for the general case of three-dimensional diffusion, but in most accurate diffusion measurements, the flow is restricted to one direction, so that 3 can be replaced by ∂/∂x where x is a Cartesian coordinate. The most common experimental situation in diffusion measurements involves gravitational stability achieved by layering of a less dense and generally less concentrated solution over a denser and generally more concentrated one. For this case, diffusive flows usually but not always move upward from higher to lower concentrations. As noted above, the typically small concentration differences used in the diffusion experiments yield diffusion coefficients and flows in the volume-fixed reference frame. Equation 2 then takes the specific form

( )

n-1

∑ (Dij)V

(Ji)V ) -

j)1

∂Cj ∂x

and for a three-component system

( ) ( )

(3)

( ) ( )

∂C1 - (D12)V ∂x ∂C1 (J2)V ) -(D21)V - (D22)V ∂x (J1)V ) -(D11)V

∂C2 ∂x ∂C2 ∂x

(4a) (4b)

A fundamental analysis of diffusive transport, based on irreversible thermodynamics,4 indicates that the driving forces for diffusion are not the concentration gradients, as assumed in Fick’s phenomenological laws, but rather the gradients of the partial molar Gibbs energy (chemical potential), (∂Gj/∂x).4-9 The simplest representation of diffusion based on these chemical-potential gradients is in terms of solvent-fixed flows and diffusion Onsager coefficients.4 On the solvent-fixed frame,

( )

n-1

(Ji)0 ) -

∑ (Lij)0 j)1

∂Gj ∂x

p,T

(5)

where Gj is the partial molar Gibbs energy (chemical potential) of solute component j in the mixture (sometimes denoted as µj, especially in the older literature), (Ji)0 is the solvent-fixed flow, and (Lij)0 is called a thermodynamic diffusion coefficient or diffusion Onsager coefficient. When eq 2 is written in terms of solvent-fixed flows and diffusion coefficients and compared with eq 5, then n-1

(Dij)0 )

∑ (Lik)0 µij

(6)

k)1

where for brevity, and in common with the published literature, we denote molarity chemical-potential derivatives as

µij )

( ) ( ) ∂Gi ∂Cj

)

p,T

∂µi ∂Cj

(7) p,T

For a three-component system, there are four such derivatives for the solutes, µ11, µ12, µ21, and µ22, where in general µ12 * µ21.10 (Lij)0 are obtained from the solution of eq 6 (four such equations for ternary systems)10 or by matrix inversion. The partial molar Gibbs energy of a solute is related to the logarithm of its activity coefficient, and accurate activity coefficient models for mixtures are usually based on the molality composition scale rather than the molarity. Thus, the derivatives

needed for eq 6 are generally evaluated with respect to the solute molalities and then multiplied by the appropriate derivatives of the molality with respect to the molarity (see below for the equations). Experimental cross-term diffusion coefficients are, in general, not equal and can be large, as found, for example, for D12 in the NaCl + MgCl2 + H2O system at 298.15 K for certain solute ratios.11 These cross-term coefficients may even have opposite signs, as is observed, for example, at certain compositions of the KCl + ZnCl2 + H2O system at 298.15 K.12 However, when the diffusion Onsager coefficients are expressed in the proper reference frame, then their cross-term coefficients should obey the Onsager reciprocal relation (ORR). For the solvent-fixed flows expressed by eq 5, the ORR is

(L12)0 ) (L21)0

(8a)

The experimental diffusion coefficients on the volume-fixed reference frame, (Dij)V, lead to a type of diffusion Onsager coefficient on the volume-fixed reference frame, but they do not obey the ORR when the driving force is defined as in eq 5. However, by transformation of the driving force to be consistent with the invariance of entropy production,4,10 the resulting volume-fixed (Lij)V do obey the ORR,4,14

(L12)V ) (L21)V

(8b)

although relationships between (Dij)V and (Lij)V are more complicated than those given by eq 6.10 (See the Appendix.) Numerous tests have been made of the ORR relation based on eq 8b for aqueous electrolyte and electrolyte + nonelectrolyte mixtures.10,13-15 The ORR has generally been found to be obeyed within realistic uncertainty limits. However, the uncertainties for these tests can be fairly large mainly because the chemical-potential concentration derivatives µij calculated from eq 7 can have large uncertainties. In the past, on the basis of examination of the chemical-potential derivatives calculated from different fits of comparable quality to a ternary-solution activity data set, we have found that the errors in the chemicalpotential derivatives are commonly about 5 to 10 % of their values. The error estimates for the present system will be discussed below. Uncertainties in the diffusion coefficients can also make significant contributions to uncertainties in the (Lij)0 and (Lij)V values. Diffusion coefficients (Dij)V measured with the Gosting diffusiometer,16 the world’s finest optical interferometer for diffusion measurements, were previously assigned uncertainties described by an earlier “rule-of-thumb” as roughly 4 times the statistical errors calculated by the propagation-of-errors method,11 and those measured with other methods or other diffusiometers will likely have even larger uncertainties. This rule of thumb was based on earlier work with 10 to 12 diffusion patterns recorded on glass photographic plates. A much greater amount of data per experiment is collected with this now automated diffusiometer, as described in the next paragraph, and the uncertainties will be discussed below. The NaCl + Na2SO4 + H2O system at 298.15 K has the potential of giving the most accurate tests of the ORR. We have reported an extensive series of diffusion coefficient measurements for this system17-21 using the Gosting diffusiometer after it had been moved to its present location at Texas Christian University. Two compositions were studied with Gouy interferometry with the diffusion patterns recorded on glass photographic plates. The other 21 compositions were studied with Rayleigh interferometry after data collection was automated with a computer-controlled scanner using a 6 cm linear diode array.

638 Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 Table 1. Values of the Experimental Volume-Fixed Diffusion Coefficients (Dij)V at T ) 298.15 K at Each of the Mean Concentrations Used in the Diffusion Studiesa j 1〉 /mol · dm-3 j 2〉 /mol · dm-3 z1 〈C 〈C 109(D11)V/m2 · s-1 109(D12)V/m2 · s-1 109(D21)V/m2 · s-1 109(D22)V/m2 · s-1 j T〉 ) 0.5 mol · dm-3 〈C 1 0.49998 0 1.4747 ( 0.001 0.166 ( 0.015 0 0.906 ( 0.015 0.95 0.47493 0.02499 (1)b 1.5018 ( 0.0005 0.1723 ( 0.0011 -0.0190 ( 0.0002 0.8873 ( 0.0004 0.1874 ( 0.0018 -0.03985 ( 0.0004 0.8713 ( 0.0007 0.90 0.45002 0.05000 (2) 1.53175 ( 0.0011 0.75 0.37499 0.12500 (3) 1.5785 ( 0.0011 0.1653 ( 0.0017 -0.0802 ( 0.0004 0.8473 ( 0.0007 0.50 0.25002 0.24998 (4) 1.6250 ( 0.0073 0.1390 ( 0.0086 -0.1312 ( 0.0027 0.8125 ( 0.0032 0.25 0.12499 0.375015 (5) 1.6591 ( 0.0072 0.0698 ( 0.0083 -0.1705 ( 0.0027 0.7997 ( 0.0031 0 0 0.50000 1.681 ( 0.002 0 -0.195 ( 0.006 0.7932 ( 0.001 0 0 0.50000 1.681 ( 0.002 0 -0.195 ( 0.006 0.7937 ( 0.001 j T〉 ) 1.0 mol · dm-3 〈C 1 1.00138 0 1.4822 ( 0.001 0.197 ( 0.015 0 0.858 ( 0.015 0.95 0.94960 0.04997 (6) 1.4873 ( 0.0004 0.2114 ( 0.0006 -0.0148 ( 0.00015 0.8302 ( 0.0002 1.5117 ( 0.0005 0.2324 ( 0.0008 -0.0349 ( 0.0002 0.8032 ( 0.0003 0.90 0.900255 0.099995 (7) 0.75 0.74990 0.24940 (8) 1.5373 ( 0.0034 0.2372 ( 0.0051 -0.0758 ( 0.0014 0.7486 ( 0.0020 0.50 0.49999 0.49996 (9) 1.5259 ( 0.0005 0.1922 ( 0.0007 -0.1158 ( 0.0002 0.6911 ( 0.0003 0.25 0.25000 0.74998 (10) 1.4726 ( 0.0011 0.1107 ( 0.0016 -0.1305 ( 0.0005 0.6616 ( 0.0006 0 0 0.99999 1.421 ( 0.027 0 -0.131 ( 0.021 0.6545 ( 0.001 0 0 1.00002 1.421 ( 0.027 0 -0.131 ( 0.021 0.6546 ( 0.001 j T〉 ) 1.5 mol · dm-3 〈C 1 1.50002 0 1.4978 ( 0.001 0.277 ( 0.015 0 0.808 ( 0.015 -0.0170 ( 0.0003 0.77595 ( 0.0005 0.95 1.424935 0.07498 (11) 1.5037 ( 0.0008 0.28825 ( 0.0013 0.90 1.34911 0.14990 (12) 1.5018 ( 0.0007 0.2970 ( 0.0011 -0.0305 ( 0.0003 0.7444 ( 0.0004 0.75 1.12447 0.37497 (13) 1.4964 ( 0.0012 0.3039 ( 0.0017 -0.0655 ( 0.0005 0.6737 ( 0.0007 0.50 0.74995 0.74996 (14) 1.4278 ( 0.0012 0.2603 ( 0.0017 -0.0917 ( 0.0005 0.5993 ( 0.0006 0.25 0.37505 1.12516 (15) 1.2953 ( 0.0008 0.1372 ( 0.0011 -0.0809 ( 0.0003 0.5738 ( 0.0004 0 0 1.50007 1.145 ( 0.02 0 -0.03 ( 0.02 0.5712 ( 0.001 0 0 1.50001 1.145 ( 0.02 0 -0.03 ( 0.02 0.5706 ( 0.001 -3 j 〈CT〉 ) 2.0 mol · dm 1 1.99994 0 1.5182 ( 0.001 0.338 0 0.763 ( 0.015 0.95 1.89909 0.09994 (16) 1.5021 ( 0.0013 0.3676 ( 0.0020 -0.0122 ( 0.0005 0.7256 ( 0.0007 0.90 1.80007 0.19998 (17) 1.4973 ( 0.0011 0.3976 ( 0.0016 -0.0259 ( 0.0005 0.6878 ( 0.0006 j T〉 ) 3.0 mol · dm-3 〈C 1 3.00019 0 1.5586 ( 0.001 0.591 0 0.666 ( 0.015 0.95 2.84561 0.14976 (18) 1.5280 ( 0.0005 0.5999 ( 0.0008 -0.0087 ( 0.0002 0.6275 ( 0.00025 0.90 2.70025 0.30004 (19) 1.5005 ( 0.0007 0.6089 ( 0.0013 -0.0160 ( 0.0003 0.5889 ( 0.0003 j T〉 ) 4.0 mol · dm-3 〈C 1 4.000175 0 1.5868 ( 0.001 0.866 0 0.572 ( 0.015 0.8299 ( 0.00165 0.5407 ( 0.0003 0.95 3.789445 0.19943 (20) 1.53705 ( 0.0007 -0.00265 ( 0.0003 0.90 3.600915 0.40010 (21) 1.4770 ( 0.0012 0.7938 ( 0.0017 -0.0005 ( 0.0005 0.5099 ( 0.0005 j T〉 ) 5.0 mol · dm-3 〈C 1 4.99938 0 1.5834 ( 0.001 1.019 0 0.500 ( 0.015 0.95 4.72944 0.24890 (22) 1.5086 ( 0.0007 0.9941 ( 0.0009 0.0048 ( 0.0003 0.4686 ( 0.0003 0.90 4.50638 0.50070 (23) 1.4218 ( 0.0007 0.9691 ( 0.0025 0.0126 ( 0.0003 0.4370 ( 0.0003 a The values of (Dij)V of the mixtures and their standard errors (from propagation-of-errors calculations) were taken from refs 17 to 21. For the limiting case as z1 f 1, (D11)V ) DV of NaCl(aq) at the same molar concentration, (D21)V ) 0, (D12)V is an extrapolated value and (D22)V is an extrapolated value equal to the trace diffusion coefficient of SO42- in NaCl(aq). Similarly, for the limiting case as z1 f 0, (D22)V ) DV of Na2SO4(aq) at the same molar concentration, (D12)V ) 0, (D21)V is an extrapolated value and (D11)V is an extrapolated value equal to the trace diffusion coefficient of the Cl- ion in Na2SO4(aq). b The numbers given in boldface font are numerical designators that are used in the subsequent tables.

This automated data acquisition yields improved precision in the diffusion coefficients because of the considerably larger amount of data that is acquired during each experiment, equivalent to photographic diffusion patterns taken at 50 different times and at more fringe positions for each time. We earlier reported extensive diffusion measurements for the limiting binary solutions NaCl + H2O and Na2SO4 + H2O at 298.15 K with a fractional uncertainty of about ( 0.1 to 0.2 % using a different diffusiometer at Lawrence Livermore National Laboratory.22,23 However, during the course of the mixture studies,17-21 we made additional measurements (seven for NaCl and nine for Na2SO4) for the binary solutions with a greater fractional precision (reproducibility) of ( 0.06 %. Extensive density data were also reported for the same NaCl + Na2SO4 + H2O mixtures,17-21 which allowed us to calculate j i of each component. These V j i values, the partial molar volumes V in turn, were used with the experimental (Dij)V to derive the corresponding solvent-fixed (Dij)0 values.

The diffusion coefficients and densities of the NaCl + Na2SO4 + H2O mixtures17-21 were measured at total mean concentraj T〉 ) 〈C j 1〉 + 〈C j 2〉 ) (0.500, 1.000, and 1.500) tions of 〈C -3 mol · dm at NaCl molarity fractions of z1 ) {1 (binary NaCl solutions), 0.95, 0.90, 0.75, 0.50, 0.25, and 0 (binary Na2SO4 j T〉 ) 〈C j 1〉 + 〈C j 2〉 ) solutions)}, along with measurements at 〈C (2.0, 3.0, 4.0, and 5.0) mol · dm-3 at NaCl molarity fractions of z1 ) 1, 0.95, and 0.90 (the higher concentration measurements20,21 could not be extended over the full range of z1 because of Na2SO4 solubility limitations). These diffusion coefficients at j T〉 were graphically extrapolated to z1 ) 1 to each constant 〈C yield the limiting value of (D12)V and of (D22)V (the sulfate trace diffusion coefficient in a NaCl solution) and correspondingly extrapolated to z1 ) 0 to yield the limiting value of (D21)V and of (D11)V (the chloride trace diffusion coefficient in a Na2SO4 solution). Because we have determined the values of (Dij)V and j T〉 and z1, we can use cross plotting (Dij)0 as functions of both 〈C

Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 639 Table 2. Values of the Derived Solvent-Fixed Diffusion Coefficients (Dij)0 at T ) 298.15 K at Each of the Mean Concentrations Used in the Diffusion Studiesa j 1〉 /mol · dm-3 j 2〉 /mol · dm-3 z1 〈C 〈C 109(D11)0/m2 · s-1 109(D12)0/m2 · s-1 109(D21)0/m2 · s-1 109(D22)0/m2 · s-1 j T〉 ) 0.5 mol · dm-3 〈C 1 0.49998 0 1.4885 ( 0.001 0.176 ( 0.015 0 0.906 ( 0.015 0.95 0.47493 0.02499 1.5151 0.1820 -0.0183 0.8878 0.90 0.45002 0.05000 1.5445 0.1970 -0.0384 0.8724 0.75 0.37499 0.12500 1.5894 0.1731 -0.0766 0.8499 0.50 0.25002 0.24998 1.6324 0.1443 -0.1238 0.8178 0.25 0.12499 0.375015 1.6629 0.0722 -0.1591 0.8069 0 0 0.50000 1.681 ( 0.002 0 -0.180 ( 0.006 0.8030 ( 0.001 0 0 0.50000 1.681 ( 0.002 0 -0.180 ( 0.006 0.8035 ( 0.001 j T〉 ) 1.0 mol · dm-3 〈C 1 1.00138 0 1.5118 ( 0.001 0.221 ( 0.015 0 0.858 ( 0.015 0.95 0.94960 0.04997 1.5152 0.2348 -0.0134 0.8315 0.90 0.900255 0.099995 1.5388 0.2540 -0.0319 0.8056 0.75 0.74990 0.24940 1.5599 0.2551 -0.0683 0.7545 0.50 0.49999 0.49996 1.5412 0.2036 -0.1005 0.7025 0.25 0.25000 0.74998 1.4801 0.1162 -0.1081 0.6779 0 0 0.99999 1.421 ( 0.027 0 -0.102 ( 0.021 0.6747 ( 0.001 0 0 1.00002 1.421 ( 0.027 0 -0.102 ( 0.021 0.6748 ( 0.001 j T〉 ) 1.5 mol · dm-3 〈C 1 1.50002 0 1.5448 ( 0.001 0.318 ( 0.015 0 0.808 ( 0.015 0.95 1.424935 0.07498 1.5485 0.3266 -0.0147 0.7780 0.90 1.34911 0.14990 1.5441 0.3330 -0.0258 0.7484 0.75 1.12447 0.37497 1.5317 0.3334 -0.0537 0.6835 0.50 0.74995 0.74996 1.4507 0.2790 -0.0688 0.6180 0.25 0.37505 1.12516 1.3062 0.1457 -0.0481 0.5994 0 0 1.50007 1.145 ( 0.02 0 -0.01 ( 0.02 0.6018 ( 0.001 0 0 1.50001 1.145 ( 0.02 0 -0.01 ( 0.02 0.6012 ( 0.001 j T〉 ) 2.0 mol · dm-3 〈C 1 1.99994 0 1.5884 ( 0.001 0.396 0 0.763 ( 0.015 0.95 1.89909 0.09994 1.5646 0.4235 -0.00895 0.7285 0.90 1.80007 0.19998 1.5563 0.4511 -0.0193 0.69375 j T〉 ) 3.0 mol · dm-3 〈C 1 3.00019 0 1.6682 ( 0.001 0.700 0 0.666 ( 0.015 0.95 2.84561 0.14976 1.6307 0.7023 -0.0033 0.6329 0.90 2.70025 0.30004 1.5968 0.7049 -0.0053 0.5996 j T〉 ) 4.0 mol · dm-3 〈C 1 4.000175 0 1.7457 ( 0.001 1.038 0 0.572 ( 0.015 0.95 3.789445 0.19943 1.68515 0.9898 0.0051 0.5491 0.90 3.600915 0.40010 1.6149 0.9422 0.0148 0.5264 j T〉 ) 5.0 mol · dm-3 〈C 1 4.99938 0 1.7912 ( 0.001 1.258 0 0.500 ( 0.015 0.95 4.72944 0.24890 1.7027 1.2183 0.0150 0.4804 0.90 4.50638 0.50070 1.60295 1.1798 0.0328 0.46035 a The values of (Dij)0 of the mixtures. For the limiting case as z1 f 1, (D11)0 ) D0 of NaCl(aq) at the same molar concentration, (D21)0 ) 0, (D12)0 is an extrapolated value and (D22)0 is an extrapolated value equal to the trace diffusion coefficient of SO42- in NaCl(aq). Similarly, for the limiting case as z1 f 0, (D22)0 ) D0 of Na2SO4(aq) at the same molar concentration, (D12)0 ) 0, (D21)0 is an extrapolated value and (D11)0 is an extrapolated value equal to the trace diffusion coefficient of the Cl- ion in Na2SO4(aq).

to assess realistic uncertainties of the individual diffusion coefficients and therefore of the derived (Lij)0 and (Lij)V values. The experimental diffusion coefficients (Dij)V from these studies and their statistical standard errors17-21 are summarized below in Table 1, as are the calculated (Dij)0 in Table 2. The µij values are also needed for the calculation of (Lij)0 and indirectly for (Lij)V. They were obtained as follows. Rard et al.24 have represented the thermodynamic activities for the NaCl + Na2SO4 + H2O system over the temperature range (278.15 to 318.15) K using an extended form of Pitzer’s ion-interaction model.25 This model includes the higher-order electrostatic term EθCl,SO4(I) for unsymmetrical mixing of electrolytes, as described in Pitzer’s Appendix B.25 However, taking the concentration derivatives of the activity coefficient equations given in ref 24 will generate highly complicated expressions for µij, in part because of the presence of this E θCl,SO4(I) term. On the other hand, Scatchard’s neutral-electrolyte equation26 is capable of representing thermodynamic activity data for

ternary electrolyte mixtures very accurately, and general expressions for the four chemical-potential molality derivatives have been recently reported by Miller.27 Thus, in this report we evaluate the neutral-electrolyte model parameters for the NaCl + Na2SO4 + H2O system at 298.15 K. We then use the resulting chemical-potential molality derivatives calculated from this model (which are related to the corresponding activity coefficient j i to calculate µij. These molality derivatives), together with V µij, together with the experimental (Dij)V and derived (Dij)0 values, allowed us to calculate the (Lij)V and (Lij)0 values at all 23 compositions for which we previously reported diffusion coefficients and density data.17-21 Uncertainties were assigned to the diffusion coefficients and activity coefficient derivatives. We then test the validity of the ORR at each composition using an expression10,14 (eq 28 below), along with the experimental (Dij)V, the activity coefficient derivatives, and their uncertainties, that yields a more direct test of the ORR10 and minimizes the accumulation of errors from including unessential µij terms that do not affect the test.

640 Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009

The goal of this analysis is to quantitatively test the ORR for ternary-solution diffusion using the extensive accurate diffusion coefficients and activity coefficient derivatives for the NaCl + Na2SO4 + H2O system at 298.15 K for which realistic estimated uncertainties have been assigned. The calculations leave no doubt as to the validity of the ORR for this system over the investigated wide composition range.

Equations Used To Represent the Osmotic and Activity Coefficients of NaCl(aq), Na2SO4(aq) and Their Mixtures Scatchard’s neutral-electrolyte model26 for a common-ion ternary mixture can be rewritten in the form

φS )

ν1m1φo1 2

∑

νimi

i)1

+

ν2m2φo2 2

∑

I

+

νimi

i)1

2

∑

(

φ ) 1 - |zMzX|Aφ

(

)

4νM2νXzM 2 (0) (1) m {CM,X + CM,X exp(-ω1,M,XI1⁄2)} (10) ν

where Aφ is the Debye-Hu¨ckel limiting law slope for water; M denotes the cation and X the anion; zM and zX (with sign) are respectively the valences of the anion and cation; νM and νX are the stoichiometric ionization numbers of the anion and cation, respectively; ν ) νM + νX; b ) 1.2 mol-1/2 · kg1/2 is (i) (i) fixed for all aqueous electrolytes; and βM,X and CM,X are fitted empirical parameters that are determined using experimental data. The corresponding equation for the molality-based mean activity coefficient is

(

νimi

i)1

b03I ) + y1y2(y1 - y2)(b12I2 + b13I3) +

∑

{(

{ (

)[

1 - 1 + R1,M,XI1⁄2 -

[y1y2(b01I + b02I + b03I ) + 2

3

νimi

i)1 y1y2(y1 - y2)(b12I2 + b13I3) + y1y2(y1 - y2)2b23I3]

(9)

where φo1 and φo2 are the osmotic coefficients of the singleelectrolyte solutions evaluated at the total stoichiometric ionic strength of the mixture; ν1m1 and ν2m2 are the ionic molalities (“osmolalities”) of the two electrolytes assuming complete dissociation; I is the molality-based stoichiometric ionic strength of the mixture; y1 and y2 are the ionic strength fractions of the two electrolytes; and h1 and h2 are the ionic molality fractions (“osmolality fractions”) of the electrolytes. Although Scatchard26 chose a particular extended DebyeHu¨ckel equation to represent φoi , there is nothing inherent in his approach that restricts it to his binary-solution equation. Pavic´evic´ et al.,28 for example, represented their isopiestic results for mixtures with eq 9 while representing the binary solution φoi contributions with the standard form of Pitzer’s ioninteraction model,25 and Miladinovic´ et al.29 used an extended form of this Pitzer model described by Archer30 for φoi . In these two studies,28,29 the hybrid form of Scatchard’s model using the ion-interaction binary-solution contributions gave a more accurate represention of the experimental osmotic coefficients than Pitzer’s standard model for electrolyte mixtures. Consequently, we chose the hybrid model to represent the activity data for NaCl + Na2SO4 + H2O. Archer30 and Rard et al.24 used an extended form of Pitzer’s ion-interaction model to represent the osmotic and activity coefficients of NaCl(aq) over a wide temperature range and of Na2SO4(aq) from T ) (273.15 to 323.15) K, respectively. Their models represent these binary-solution thermodynamic properties very accurately and are accepted here. This extended ioninteraction equation for the osmotic coefficient of a solution of an electrolyte of arbitrary valence type can be written in the general form

(

)

)

()

( )

)

}]

R1,M,X2I exp(-R1,M,XI1⁄2) 2

[ ( )

(1) 2νM2νXzM 2 4CM,X (0) m 3CM,X + × ν ωM,X4I2

{ {

}

I1⁄2 2 + ln(1 + bI1⁄2) + 1⁄2 b 1 + bI (1) 2νMνX 2βM,X (0) m 2βM,X + × ν R1,M,X2I

ln γ( ) -|zMzX|Aφ

[y1y2(b01I + b02I2 +

3

y1y2(y1 - y2)2b23I3] I ) h1φo1 + h2φo2 + 2

)( )

2νMνX I1⁄2 (0) + m{βM,X + 1⁄2 ν 1 + bI (1) βM,X exp(-R1,M,XI1⁄2)} +

6 - 6 + 6ωM,XI1⁄2 + 3ωM,X2I + ωM,X3I3⁄2 -

(

ωM,X4I2 2

)}

+

}]

exp(-ωM,XI1⁄2)

(11)

Table 3 lists the parameters of this model for NaCl(aq) and Na2SO4(aq) at 298.15 K. The parameters for Na2SO4(aq) were taken from the third column of Table 3 of Rard et al.24 and those for NaCl(aq) were calculated from the equations and parameters reported by Archer.30 The Debye-Hu¨ckel limiting law slope for water at T ) 298.15 K and p ) 0.1 MPa, Aφ ) 0.391 475 mol-1/2 · kg1/2, was calculated from the Archer and Wang evaluation.31 Rard et al.24 based their temperature-dependent model for NaCl + Na2SO4 + H2O on isopiestic data, Emfs from various types of cells, and enthalpies of mixing, whose data sources and assigned weights are summarized in their Table 2. However, the available isopiestic results were mainly restricted to 298.15 K; many of the Emf studies involved either sodium-ionresponsive glass electrodes or a so-called ion-selective electrode (none of which show a truly Nernstian response), and the enthalpies of mixing (although probably reliable) are mostly restricted to a few selected ionic strengths or equivalent molalities. The most extensive isopiestic data for the NaCl + Na2SO4 + H2O system were measured at 298.15 K, and this is the temperature at which we need to calculate values of µij. We thus restricted the evaluation of the neutral-electrolyte model parameters to this temperature. There are four sets of isopiestic measurements at 298.15 K: 15 values from Wu et al.,32 4 from Robinson et al.,33 33 from Filippov and Cheremnykh,34 and 119 values of Platford that are listed in Table 1 of Rard et al..24 This last set is the most extensive and spans the full composition range from the dilute into the oversaturated (supersaturated) concentration region, I ) (0.176 to 10.050) mol · kg-1. The

Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 641 Table 3. Parameters for the Extended Ion-Interaction (Pitzer) Models for Na2SO4(aq) and NaCl(aq) and the Debye-Hu¨ckel Limiting Law Slope, All at T ) 298.15 K parameter valuea parameter

Na2SO4(aq)

NaCl(aq)

(0) βM,X /mol-1 · kg (1) βM,X /mol-1 · kg (0) CM,X /mol-2 · kg2 (1) CM,X /mol-2 · kg2 RM,X/mol-1/2 · kg1/2 ωM,X/mol-1/2 · kg1/2 Aφ/mol-1/2 · kg1/2 Imax/mol · kg-1 s(φ)

9.1423 · 10-3 1.001 50 2.5960 · 10-3 0.162 097 2.0 2.25 0.391 475 11.44 0.001 00

0.080 634 21 0.263 097 73 2.6239 · 10-4 -0.010 051 73 2.0 2.5 0.391 475 6.15

a Parameters for Na2SO4(aq) were taken from Table 3 of Rard et al.,24 and those for NaCl(aq) were calculated from the temperature and pressure coefficients reported by Archer.30 The Debye-Hu¨ckel limiting law slope Aφ is from Archer and Wang.31 Imax is the maximum ionic strength of the source data used for evaluation of the model parameters {the saturated solution for NaCl(aq) and oversaturated (supersaturated) solution for Na2SO4(aq)}.

deviations as a function of the ionic strength. Two of the threeparameter fits seem to be the best with essentially random deviations, and using four or more parameters gives no improvement in the RMSE(φ) value but merely results in larger standard coefficient errors. On the basis of RMSE(φ) and the standard coefficient errors, the two best (optimal) representations of the experimental osmotic coefficients are those involving two symmetrical mixing parameters and one asymmetrical one, {b01, b02, b13} and {b01, b02, b23}, with {b01, b02, b23} having slightly better coefficient errors. The latter coefficient set is our recommended fit. Figure 1 is a deviation plot of φ(model) φ(expt) against the ionic strength for the fit with the recommended {b01, b02, b23} parameter set. We note that when eq 10 is used to represent φoi of eq 9, replace m with I for NaCl(aq) and m with I/3 for Na2SO4(aq).

Calculation of the Chemical-Potential Derivatives µij and µm ij and the Concentration Derivatives (Dmi/DCj)p,T,Ck ( k+j ) Equation 7 can be rewritten as

24

analysis given by Rard et al. indicates that the osmotic coefficients of Platford24 and Robinson et al.33 are completely consistent, whereas those of Filippov and Cheremnykh34 are generally significantly lower and scattered, and those of Wu et al.32 are systematically higher with the discrepancies becoming larger as the ionic strength fraction of Na2SO4 in the mixtures becomes larger. As noted by Rard et al., the binary-solution osmotic coefficients for Na2SO4(aq) from the study of Wu et al. are also systematically high, which implies that their32 osmotic coefficients for the NaCl + Na2SO4 + H2O mixtures are probably also systematically high and slightly skewed as a function of the NaCl solute fraction z1. On the basis of the considerations described in the preceding paragraph, evaluation of the bij mixing parameters of eq 9 was based on the two consistent sets of isopiestic data,24,33 with equal weight given to each osmotic coefficient except for several points that were weighted zero by Rard et al.24 Thus, 112 osmotic coefficients were used to evaluate the bij values. For these parameter evaluations, the quantity being minimized by least squares was ∆φ ) φ(expt) - φS, where φ(expt) is an experimental osmotic coefficient and φS is an osmotic coefficient as given by Scatchard’s neutral-electrolyte model, eq 9. Table 4 gives the evaluated bij mixing parameters for 10 trial combinations of the bij, along with the ratio of the standard error of each coefficient divided by that coefficient’s value (fractional errors) and the root-mean-square error for the fit RMSE(φ). As can be seen from Table 4 and from deviation plots (not shown in most cases), the one-parameter fit is not adequate and the two-parameter fits are fairly good but have small systematic

µij )

( ) ∂Gi ∂Cj

2

) p,T

∑

k)1

( ) ( ) ∂Gi ∂mk

p,T,Cl(l*k)

∂mk ∂Cj

(12) p,T,Cl(l*j)

We first consider the first partial differential on the righthand side (RHS) of this equation and the relation between Gi and the molality and mean activity coefficient of solute component i when ideality is defined on the molality scale:

µijm )

( ) ∂Gi ∂mj

) p,T

∂ {Go + RT ln(VMiVMiVXiVXimiViγ(,iVi)}p,T ∂mj m,i (13)

where R ) 8.3145 J · K-1 · mol-1 is the gas constant and T the absolute temperature. We now consider separately the two cases when i * j and when i ) j. For the first case,

µijm ) RT

{

∂ ln γ(,i ∂ {ln(miViγ(,iVi)}p,T ) ViRT ∂mj ∂mj

}

i*j

, p,T

(14a) and for the second case

µiim ) RT

( {

∂ ln γ(,i 1 ∂ {ln(miViγ(,iVi)}p,T ) ViRT + ∂mi mi ∂mi

}) p,T

(14b) Miller27 has recently reported explicit equations for {∂ ln γ(,i/ ∂mi }p,T and {∂ ln γ(,i/∂mj }p,T (i * j) when Scatchard’s neutralelectrolyte model, eq 9, is used to represent the osmotic and

Table 4. Mixing Parameter Values of Scatchard’s Neutral-Electrolyte Equation for NaCl + Na2SO4 Aqueous Mixtures at T ) 298.15 K, Using the Extended Ion-Interaction (Pitzer) Model Parameters for the Single Electrolytes Na2SO4(aq) and NaCl(aq) Reported in Table 3a b01/kg · mol-1 -0.049 872 1 (0.0115) -0.066 949 1 (0.0089) -0.058 607 7 (0.0084) -0.071 488 8 (0.0162) -0.068 940 6 (0.0097) -0.069 849 1 (0.0103) -0.069 153 5 (0.0092) -0.071 009 9 (0.0158) -0.069 810 4 (0.0172) -0.070 784 3 (0.0227) a

b02/kg2 · mol-2

b03/kg3 · mol-3

0.003 055 1 (0.0329) 0.000 251 5 (0.0479) 0.004 879 6 (0.0861) -0.000 162 9 (0.2246) 0.003 645 4 (0.0407) 0.003 818 6 (0.0412) 0.003 616 9 (0.0358) 0.004 534 8 (0.0923) -0.000 093 9 (0.4410) 0.003 799 3 (0.1331) 0.000 002 4 (24.90) 0.004 690 4 (0.1934) -0.000 138 7 (0.8711)

b12/kg2 · mol-2

b13/kg3 · mol-3

b23/kg3 · mol-3

RMSE(φ)

0.002 95 0.000 97 0.001 33 0.000 89 0.000 585 5 (0.1988) 0.000 88 0.000 085 4 (0.1706) 0.000 85 -0.000 087 6 (0.1695) 0.000 85 0.000 424 6 (0.3168) 0.000 86 0.000 086 2 (0.2863) 0.000 85 0.001 056 4 (0.6767) -0.000 209 7 (0.8454) -0.000 138 9 (0.5137) 0.000 84

To the right of each parameter value, in parentheses, is given the ratio of the standard error of the bij coefficient divided by that coefficient (fractional error). RMSE(φ) is the root-mean-square error of the fit to the experimental osmotic coefficients.

642 Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009

{

o ∂ ln γ(,i ∂I

(

{

}

T,p

{

) -|zMzX|Aφ

(

(0) (1) βM,X + βM,X 1-

8VM2VX V3|zMzX2|

)

}(

)

8VMVX 1.5I-1⁄2 + b + 2 × 1⁄2 2 (1 + bI ) V |zMzX|

)

}

R1,M,XI1⁄2 exp(-R1,M,XI1⁄2) + 4

(0) (1) {6CM,X I + CM,X (6I - ωM,XI3⁄2) exp(-ωM,XI1⁄2)}

(17)

The remaining derivative on the RHS of eq 12 is the molality/ molarity concentration derivative (∂mi/∂Cj)p,T,Ck(k+j). Miller10 gave equations for evaluating this derivative:

( ) ∂mi ∂Cj

) p,T,Ck(k*j)

)

Figure 1. Deviations between the optimized Scatchard model values from eq 9 and the experimental osmotic coefficients, φ(fit) - φ(expt), as a function of the molality-based ionic strength I for the recommended {b01, b02, b23} parameter set of Table 4 and the binary-solution parameters of Table 3.

activity coefficients of ternary electrolyte solutions. This is the same model that we are using so we need not repeat those equations here. However, Miller’s equations contain the derivative {∂ ln γo(,i/∂I }p,T, where γo(,i is the mean activity coefficient of electrolyte i in its binary solution at the total ionic strength I of the ternary mixture. This ionic strength derivative expression has not been reported previously for the extended ion-interaction (Pitzer) model used here, eq 11, and is therefore derived here. Equation 11 as written is a function of both the stoichiometric ionic strength and the molality of electrolyte i, and it is more convenient to express it solely in terms of the ionic strength before taking the ionic strength derivative {∂ ln γo(,i/∂I }p,T. The molality and ionic strength of an electrolyte of arbitrary valence type are related by

m)

2I V|zMzX|

(15)

Inserting eq 15 into eq 11 yields

{(

)

(

}

I1⁄2 2 + ln(1 + bI1⁄2) + 1⁄2 b 1 + bI (1) βM,X 8VMVX (0) β I + 1 - 1 + R1,M,XI1⁄2 M,X 2 2 V |zMzX| R1,M,X

o ln γ(,i ) -|zMzX|Aφ

{ {

)[

(

)

()

( ){ (

}]

R1,M,X2I exp(-R1,M,XI1⁄2) 2 8VM2VX V3|zMzX2|

6 - 6 + 6ωM,XI

1⁄2

)[

(0) 2 3CM,X I +

( ) ωM,X4

+ 3ωM,X2I + ωM,X3I3⁄2 -

( )[ ( )

]

jj CiV 1000 δij + C0M0 j0 C0V 1000 0V A C0M0 ij

] (18)

( ) ( ) ∂G1 ∂m2

)

p,T

∂G2 ∂m1

) µm21

(19)

p,T

Calculation of the (Dij)0, (Lij)0, µij, µm ij , and (Lij)V Values and of Tests of the ORR ×

(

ωM,X4I2 2

)}

}]

exp(-ωM,XI1⁄2) The ionic strength derivative is then given by

jj CiV mi δij + Cj j0 C0V

j j is the partial molar volume of solute j in the mixed where V j 0 is the partial molar volume of the electrolyte solution and V solvent (all in units of cm3 · mol-1), δij ) 1 if i ) j and δij ) 0 if i * j (δij is the Kronecker δ), and A0V ij is defined by eq 18 and also in the Appendix. Four experiments were performed at each overall composition in the diffusion studies for the NaCl + Na2SO4 + H2O system,17-21 each of which involved a pair of solutions, and thus eight density measurements were made. Also reported in these studies were the overall average molarities of both solute components for the four experiments, with the molalities corresponding to these average molarities and the partial molar volumes of the solutes and water at these overall average molarities. Therefore, all of the information needed to calculate the values of (∂mi/∂Cj)p,T,Ck(k+j) are already available. Table 5 reports the derived values of µ11/RT, µ12/RT, µ21/ RT, and µ22/RT. They were evaluated by using the selected {b01, b02, b23} parameter set of Table 4 to calculate the values of µm ij ) (∂Gi/∂mj)p,T from eqs 14a and 14b, followed by the calculation of (∂mi/∂Cj)p,T,Ck(k+j) by insertion of eq 18 into eq 12. The molality activity coefficient derivatives of eqs 14a and 14b are reported in Table 6. It should be noted that µ12 * µ21 for the cross-term chemical-potential derivatives with respect to the molarity whereas for any thermodynamically consistent molality-based model for ternary electrolyte solutions,

µm12 )

+

(1) 4CM,X

)

( )[

×

The Appropriate Equations for Ternary Systems. The equations below14 can be obtained most easily by matrix methods, as shown in the Appendix. (Dij)0 are obtained from (Dij)V from14 (also see the Appendix)

(16)

(Dij)0 ) where

2

2

k)1

k)1

∑ Rki(Dkj)V ) ∑ Aik0V(Dkj)V

(20)

Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 643

ji CjV Rij ) δij + ) Aji0V j C0V0

(L11)0 ) 10-3[(D11)0 µ22 - (D12)0 µ21] ⁄ S

(21)

-3

(See the Appendix for the meaning of A˜0V ji , which is the transpose of A0V ij .) We also note that

µij )

(

jj CkV 1 µikm δkj + C0M0 k)1 j0 C0V 2

∑

)

)

1 µmA0V C0M0 k)1 ik kj

2

∑

(22a)

{ ( ) ( )} { ( ) ( )}

j2 j2 C2V C1V 1 µ12 ) µm11 + µm12 1 + C0M0 j0 j0 C0V C0V

(22b)

j1 j1 C1V C2V 1+ + µm22 j0 j0 C0V C0V

(22c)

1 µm C0M0 21

(L21)0 ) 10-3[(D21)0 µ22 - (D22)0 µ21] ⁄ S

(23c)

(L22)0 ) 10-3[(D22)0 µ11 - (D21)0 µ12] ⁄ S

(23d)

S ) µ11µ22 - µ12µ21

(24)

The factor 10 appears in these equations because of SI units of m2 · s-1 for (Dij)0 and mol · dm-3 for molar concentrations. Similarly, expressions for (Lij)V are obtained by solving the four expressions contained in the volume-fixed analogue of eq 6 (see the Appendix). The results are10,14

where M0 is the molar mass of the solvent (H2O in our case) in units of kg · mol-1. Explicit results for the cross derivatives are

µ21 )

(23b)

-3

∑

1 µmR C0M0 k)1 ik jk

(L12)0 ) 10 [(D12)0 µ11 - (D11)0 µ12] ⁄ S

where

2

)

(23a)

Although the molality derivatives ) by eq 19, comparison of eq 22b with eq 22c shows that for the corresponding molarity derivatives µ12 * µ21. Expressions for (Lij)0 can also be obtained by solution of the four expressions contained in eq 6 or by matrix inversion of eq 6. Either way, the results are35 µm 12

(L11)V ) [(D11)V a22 - (D12)V a12] ⁄ A

(25a)

(L12)V ) [(D12)V a11 - (D11)V a21]⁄A

(25b)

(L21)V ) [(D21)V a22 - (D22)V a12] ⁄ A

(25c)

(L22)V ) [(D22)V a11 - (D21)V a21] ⁄ A

(25d)

where

aij )

µm 21

∑ 2k)1 Rjkµki

(26)

and

A ) a11a22 - a12a21

(27)

The (Dij)V, (Dij)0, µij, {∂ ln γ(,i/∂mj }p,T, (Lij)V, and (Lij)0 values respectively are contained in Tables 1, 2, and 5 to 8.

Table 5. Values of µ11/RT, µ12/RT, µ21/RT, and µ22/RT at T ) 298.15 K at Each of the Mean Concentrations Used in the Diffusion Studies,17-21 Calculated by Using the Selected {b01, b02, b23} Parameter Set of Table 4 and the Binary-Solution Parameters of Table 3a j 1〉 /mol · dm-3 j 2〉 /mol · dm-3 z1 〈C 〈C (µ11/RT)/dm3 · mol-1 (µ12/RT)/dm3 · mol-1 (µ21/RT)/dm3 · mol-1 (µ22/RT)/dm3 · mol-1 -3 j T〉 ) 0.5 mol · dm 〈C 0.95 0.47493 0.02499 3.80934 2.44097 2.44702 42.61239 0.90 0.45002 0.05000 3.87346 2.37606 2.38001 22.58730 0.75 0.37499 0.12500 4.17786 2.18170 2.18472 10.45933 0.50 0.25002 0.24998 5.32675 1.90553 1.90482 6.17388 0.25 0.12499 0.375015 9.19093 1.68573 1.68571 4.56691 j T〉 ) 1.0 mol · dm-3 〈C 0.95 0.94960 0.04997 2.02792 1.23170 1.227065 20.93400 0.90 0.900255 0.099995 2.06340 1.20661 1.20375 10.93678 0.75 0.74990 0.24940 2.22611 1.137415 1.131785 4.94385 0.50 0.49999 0.49996 2.81439 1.03247 1.02369 2.86029 0.25 0.25000 0.74998 4.75789 0.95309 0.93957 2.11226 j T〉 ) 1.5 mol · dm-3 〈C 0.95 1.424935 0.07498 1.45466 0.88163 0.872905 13.838825 0.90 1.34911 0.14990 1.48217 0.87404 0.86431 7.19708 0.75 1.12447 0.37497 1.59774 0.84599 0.83303 3.22139 0.50 0.74995 0.74996 2.00106 0.80271 0.78402 1.87872 0.25 0.37505 1.12516 3.30692 0.76988 0.74592 1.40945 j T〉 ) 2.0 mol · dm-3 〈C 0.95 1.89909 0.09994 1.18112 0.73884 0.72478 10.37087 0.90 1.80007 0.19998 1.20317 0.739105 0.72318 5.39176 j T〉 ) 3.0 mol · dm-3 〈C 0.95 2.84561 0.14976 0.92912 0.64890 0.62348 7.02213 0.90 2.70025 0.30004 0.94702 0.65832 0.630815 3.70346 j T〉 ) 4.0 mol · dm-3 〈C 0.95 3.789445 0.19943 0.82372 0.651535 0.61600 5.45450 0.90 3.600915 0.40010 0.84063 0.66991 0.63075 2.96678 -3 j 〈CT〉 ) 5.0 mol · dm 0.95 4.72944 0.24890 0.777125 0.69453 0.64663 4.60689 0.90 4.50638 0.50070 0.79544 0.721035 0.67090 2.61550 a The values of µij/RT are given to the number of figures used for the calculation of (Lij)0 and (Lij)V. Although these values are reported to several more figures than justified by their absolute accuracy, we retained extra figures in the calculations to avoid round-off errors because various combinations of µij are involved in the calculations, e.g., (µ11µ22 - µ12µ21).

644 Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009

The (Lij)V values of Table 7 are those based on the volumefixed driving force defined to be consistent with the invariance of entropy production.4,10,14 The ORR can be tested directly by comparing L12 and L21 on either reference frame calculated from the appropriate j i, and µij values. However, equations given above, using the Dij, V the Lij expressions in terms of Dij and µij have many extraneous terms including the determinants from matrix inversions, as can be inferred from eq 6. These extraneous terms contribute to the estimated error from the propagation-of-errors equations, yet they are unessential in comparing L12 with L21. An alternative test of the ORR can eliminate or simplify some of these terms by various transformations that give necessary and sufficient conditions for the ORR. The simpler test equation10,14 is advantageous because it cancels out just those terms on both sides of that equation that would otherwise make irrelevant contributions to the estimated error. It is also desirable to use the volume-fixed (Dij)V directly for this test because they are the experimental quantities, whereas j i terms along with the solvent-fixed (Dij)0 require additional V their own (small) experimental errors. Thus, the simpler ORR test equation for a ternary system is10,14

a11(D12)V + a12(D22)V ) a21(D11)V + a22(D21)V

(28)

This test condition is derived by equating eq 25b to eq 25c and requires that the denominator of these equations be nonzero, i.e., A * 0. The RHS, left-hand side (LHS), RHS - LHS, and its uncertainty of this test equation are given in Table 9.

Estimates of the Errors in Experimental Quantities To estimate the fractional uncertainties in the activity coefficient derivatives for the selected best fit with {b01, b02, b23}, the (∂ ln γ(,i/∂mj)p,T ) (1/γ(,i)(∂ γ(,i/∂mj)p,T values were also calculated for two other sets of parameters from Table 4 for fits with low and comparable RMSE(φ): {b01, b02, b13} and {b01, b02, b03, b12, b13, b23}. The maximum differences among these three sets of calculated (∂ ln γ(,i/∂mj)p,T values at each composition, ∆(∂ ln γ(,i/∂mj)p,T, are reported in Table 6. Inspection of the ∆(∂ ln γ(,i/∂mj)p,T in Table 6 shows some obvious trends. (i) Values of all three of the ∆(∂ ln γ(,i/∂mj)p,T for a fixed z1 solute ratio continually increase with increasing j T〉. (ii) At any fixed value of 〈C j T〉, the total concentration 〈C values of ∆(∂ ln γ(,1/∂m1)p,T are smaller as z1 f 1 and larger as z1 f 0. Thus, the NaCl derivative is more accurate as the NaCl(aq) binary composition is approached and least accurate

Table 6. Values of (D ln γ(,i/Dmj)p,T at T ) 298.15 K at Each of the Mean Concentrations and the Corresponding Molalities Used in the Diffusion Studies,17-21 Calculated by Using the Selected {b01, b02, b23} Parameter Set of Table 4 and the Binary-Solution Parameters of Table 3, and Maximum Differences among Values from Three Mixing Parameter Setsa j 1,C j 2) and m1(C j 2〉 j 1〉 j 1,C j 2) 〈C (∂ ln γ(,1/∂m1)p,T m2(C 〈C V1(∂ ln γ(,1/∂m2)p,T (∂ ln γ(,2/∂m2)p,T z1

mol · dm-3

mol · dm-3

kg · mol-1

kg · mol-1

j T〉 ) 0.5 mol · dm 〈C -0.11678 (0.00006) -1.38813 (0.00212) -0.09977 (0.00014) -1.28227 (0.00227) -0.06138 (0.00033) -1.04351 (0.00271) -0.02081 (0.00070) -0.79215 (0.00341) 0.00546 (0.00108) -0.63116 (0.00407) j T〉 ) 1.0 mol · dm-3 〈C -0.00824 (0.00010) -0.70622 (0.00123) 0.00197 (0.00021) -0.64340 (0.00123) 0.02536 (0.00055) -0.50097 (0.00138) 0.05187 (0.00119) -0.34325 (0.00227) 0.07026 (0.00191) -0.23766 (0.00331) j T〉 ) 1.5 mol · dm-3 〈C 0.03584 (0.00010) -0.42975 (0.00199) 0.04370 (0.00022) -0.38247 (0.00180) 0.06259 (0.00062) -0.27065 (0.00115) 0.08476 (0.00145) -0.14440 (0.00148) 0.10096 (0.00260) -0.05742 (0.00263) j T〉 ) 2.0 mol · dm-3 〈C 0.06179 (0.00013) -0.26738 (0.00266) 0.06866 (0.00025) -0.22672 (0.00212) j T〉 ) 3.0 mol · dm-3 〈C 0.09306 (0.00037) -0.07028 (0.00370) 0.09892 (0.00066) -0.03584 (0.00251) j T〉 ) 4.0 mol · dm-3 〈C 0.11222 (0.00092) 0.05427 (0.00863) 0.11762 (0.00172) 0.08681 (0.00735) j T〉 ) 5.0 mol · dm-3 〈C 0.12566 (0.00176) 0.14576 (0.02101) 0.13090 (0.00332) 0.17907 (0.01862)

kg · mol-1

mol · kg-1

-3

0.95 0.90 0.75 0.50 0.25

0.47493 0.45002 0.37499 0.25002 0.12499

0.02499 0.05000 0.12500 0.24998 0.375015

0.95 0.90 0.75 0.50 0.25

0.94960 0.900255 0.74990 0.49999 0.25000

0.04997 0.099995 0.24940 0.49996 0.74998

0.95 0.90 0.75 0.50 0.25

1.424935 1.34911 1.12447 0.74995 0.37505

0.07498 0.14990 0.37497 0.74996 1.12516

0.95 0.90

1.89909 1.80007

0.09994 0.19998

0.95 0.90

2.84561 2.70025

0.14976 0.30004

0.95 0.90

3.789445 3.600915

0.19943 0.40010

0.95 0.90

4.72944 4.50638

0.24890 0.50070

a

-1.66658 0.00026) -1.55758 (0.00025) -1.31214 (0.00019) -1.05543 (0.00014) -0.89310 (0.00007)

0.480654 0.455455 0.379555 0.253120 0.126579

0.025296 0.050605 0.126524 0.253075 0.379785

-0.95979 (0.00052) -0.89690 (0.00047) -0.75622 (0.00037) -0.60467 (0.00023) -0.50649 (0.00014)

0.970554 0.920260 0.767013 0.512051 0.256408

0.051071 0.102217 0.255091 0.512020 0.769204

-0.676985 (0.00078) -0.63206 (0.00070) -0.528145 (0.00050) -0.41539 (0.00029) -0.34042 (0.00020)

1.471642 1.393896 1.163458 0.778110 0.390373

0.077436 0.154878 0.387972 0.778118 1.171128

-0.51349 (0.00103) -0.47715 (0.00100)

1.982979 0.104356 1.881053 0.208973

-0.31746 (0.00291) -0.29081 (0.00244)

3.041664 0.160075 2.891262 0.321261

-0.19324 (0.00550) -0.17203 (0.00466)

4.153577 0.218593 3.959282 0.439914

-0.09993 (0.00911) -0.08204 (0.00780)

5.324414 0.280213 5.098715 0.566518

The values of (∂ ln γ(,i/∂mj)p,T are given to the number of figures used for the calculation of (Lij)0 and (Lij)V. Although these values are reported to several more figures than justified by their absolute accuracy, we retained extra figures in the calculations to avoid round-off errors. There are three tabulated derivatives given for each composition. Only the one cross derivative V1 (∂ ln γ(,1/∂m2)p,T is necessary because of the equality V1 (∂ ln γ(,1/ m ∂m2)p,T ) V2 (∂ ln γ(,2/∂m1)p,T, which comes from µm 12 ) µ21. The uncertainties, ∆(∂ ln γ(,i/∂mj)p,T, given in parentheses to the right of the corresponding (∂ ln γ(,i/∂mj)p,T from the selected {b01, b02, b23} parameter set, are the maximum differences among the values calculated for three sets of parameters from Table 4 with low and comparable RMSE(φ): {b01, b02, b23}, {b01, b02, b13}, and {b01, b02, b03, b12, b13, b23}.

Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 645 Table 8. Values of the Solvent-Fixed Onsager Diffusion Coefficients (Lij)0 at T ) 298.15 K at Each of the Mean Concentrations Used in the Diffusion Studiesa

Table 7. Values of the Volume-Fixed Onsager Diffusion Coefficients (Lij)V at T ) 298.15 K at Each of the Mean Concentrations Used in the Diffusion Studiesa 1012RT(L11)V composition

b

mol · m

-1

·s

1 2 3 4 5

0.403 0.414 0.412 0.333 0.190

6 7 8 9 10

0.741 0.757 0.745 0.592 0.327

11 12 13 14 15

1.032 1.038 1.009 0.783 0.420

16 17

1.258 1.263

18 19

1.592 1.583

20 21

1.766 1.736

22 23

1.804 1.740

-1

1012RT(L12)V -1

-1

1012RT(L21)V -1

mol · m · s mol · m · s j T〉 ) 0.5 mol · dm-3 〈C -0.019 -0.019 -0.036 -0.036 -0.072 -0.069 -0.082 -0.081 -0.056 -0.055 j T〉 ) 1.0 mol · dm-3 〈C -0.035 -0.033 -0.065 -0.064 -0.129 -0.127 -0.152 -0.151 -0.099 -0.100 -3 j 〈CT〉 ) 1.5 mol · dm -0.048 -0.048 -0.091 -0.089 -0.182 -0.178 -0.209 -0.211 -0.139 -0.137 j T〉 ) 2.0 mol · dm-3 〈C -0.060 -0.057 -0.111 -0.110 j T〉 ) 3.0 mol · dm-3 〈C -0.076 -0.078 -0.143 -0.147 j T〉 ) 4.0 mol · dm-3 〈C -0.086 -0.092 -0.171 -0.170 j T〉 ) 5.0 mol · dm-3 〈C -0.099 -0.101 -0.179 -0.183

-1

1012RT(L22)V -1

mol · m

-1

·s

1012RT(L11)0 composition

b

-1

mol · m

·s

0.022 0.042 0.095 0.156 0.194

1 2 3 4 5

0.410 0.421 0.417 0.335 0.191

0.042 0.080 0.180 0.293 0.351

6 7 8 9 10

0.768 0.783 0.764 0.601 0.330

0.059 0.114 0.254 0.401 0.466

11 12 13 14 15

1.092 1.092 1.048 0.801 0.424

0.074 0.142

16 17

1.359 1.355

0.096 0.183

18 19

1.800 1.769

0.109 0.206

20 21

2.097 2.026

0.115 0.210

22 23

2.254 2.130

a R ) 8.3145 J · K-1 · mol-1 and T ) 298.15 K. These (Lij)V values are those calculated using the driving force defined to be consistent with the invariance of entropy production.4,10 b The numbers in this column denote the compositions as defined in Table 1.

as its trace composition is approached, which is an expected j T〉, the values result. Correspondingly, at any fixed value of 〈C of ∆(∂ ln γ(,2/∂m2)p,T are smaller as z1 f 0 and larger as z1 f 1. Thus, the Na2SO4 derivative is more accurate as the Na2SO4(aq) binary composition is approached and least accurate as its trace composition is approached. (iii) The uncertainty of the cross-derivative ∆(∂ ln γ(,1/∂m2)p,T increases as z1 f 0 at j T〉 ) 0.5 mol · dm-3, is initially constant and then increases 〈C j T〉 ) 1.0 mol · dm-3, but goes through a minimum as z1 f 0 at 〈C j T〉 ) 1.5 mol · dm-3 with increasing values as z1 f 0 and at 〈C z1 f 1. The corresponding fractional uncertainties are not an ideal way of describing these differences because both (∂ ln γ(,1/ ∂m1)p,T and (∂ ln γ(,1/∂m2)p,T change sign (and thus must be zero at some mixture compositions), but the absolute values of their errors are used in this paragraph because it is a convenient way of assigning uncertainties needed for the (Lij)0 and (Lij)V calculations. The values of the fractional uncertainties Fij ) {∆(∂ ln γ(,i/∂mj)p,T/(∂ ln γ(,i/∂mj)p,T} are nearly always e0.03 when j T〉 e 3.0 mol · dm-3, with the individual Fij values usually 〈C j T〉 ) 0.5 being well below 0.02. However, F11 ≈ 0.20 at 〈C j T〉 ) 1.0 mol · dm-3 when z1 ) 0.25 and F11 ≈ 0.11 at 〈C mol · dm-3 when z1 ) 0.90, although the actual errors are very small in both cases because the values of (∂ ln γ(,1/∂m1)p,T are j T〉 ) (4.0 and 5.0) mol · dm-3 the near zero. In addition, at 〈C values of Fij may exceed 0.10 at some compositions and F12 at j T〉 ) 3.0 mol · dm-3 reaches 0.07 mol · dm-3 when z1 ) 0.90. 〈C We note that the φo1 and φo2 equations24,30 used for the binarysolution contributions to eq 9 are the same for all of the

-1

1012RT(L12)0 -1

-1

1012RT(L21)0 -1

mol · m · s mol · m · s j T〉 ) 0.5 mol · dm-3 〈C -0.019 -0.019 -0.036 -0.036 -0.070 -0.068 -0.080 -0.079 -0.055 -0.053 j T〉 ) 1.0 mol · dm-3 〈C -0.034 -0.032 -0.063 -0.062 -0.124 -0.123 -0.146 -0.144 -0.094 -0.095 -3 j 〈CT〉 ) 1.5 mol · dm -0.046 -0.046 -0.086 -0.084 -0.172 -0.167 -0.194 -0.196 -0.128 -0.126 j T〉 ) 2.0 mol · dm-3 〈C -0.056 -0.053 -0.102 -0.102 j T〉 ) 3.0 mol · dm-3 〈C -0.066 -0.068 -0.124 -0.129 j T〉 ) 4.0 mol · dm-3 〈C -0.069 -0.076 -0.140 -0.139 j T〉 ) 5.0 mol · dm-3 〈C -0.075 -0.077 -0.136 -0.140

-1

1012RT(L22)0 mol · m-1 · s-1 0.022 0.042 0.095 0.157 0.196 0.042 0.081 0.181 0.298 0.364 0.059 0.114 0.256 0.413 0.494 0.074 0.143 0.096 0.185 0.110 0.209 0.116 0.215

a R ) 8.3145 J · K-1 · mol-1 and T ) 298.15 K. b The numbers in this column denote the compositions as defined in Table 1.

derivative calculations and thus do not contribute to the observed variations from using different mixing parameter combinations. However, the errors in the derivatives of those binary-solution models do contribute to the overall errors in (∂ ln γ(,i/∂mj)p,T. We have not estimated any errors of the binary-solution derivatives, although the uncertainties in the NaCl(aq) model may contribute somewhat at the highest concentrations that are near its fitting (solubility) limit. Because the φo1 equation30 used here for NaCl(aq) was also used as the reference standard for the NaCl + Na2SO4 + H2O mixtures24 and was the standard for much of the Na2SO4(aq) isopiestic data, the binary- and ternary-solution osmotic and activity coefficient equations used in our (∂ ln γ(,i/∂mj)p,T calculations should be highly consistent internally. The larger uncertainties of the (∂ ln γ(,i/∂mj)p,T values when j T〉 ) (4.0 and 5.0) mol · dm-3 are not surprising because the 〈C isopiestic data used for evaluation24 of the bij parameters of Table 4 extend only to the total molalities of mT ≈ 3.8 mol · kg-1, with only a few points above mT ≈ 3.6 mol · kg-1. However, the ionic strengths of the more concentrated isopiestic solutions do extend above those of the diffusion experiments (which are restricted to the NaCl-rich region with z1 ) 0.90 and 0.95) and provide some constraint on eq 9 at high concentrations, as do the binary-solution contributions that are valid to higher ionic strengths than the diffusion compositions. After considering the issues described in the above discussion, we conservatively assign 0.05 uncertainties to the calculated j T〉 ) 3.0 j T〉 e 3.0 mol · dm-3, except for F12 at 〈C Fij when 〈C -3 mol · dm , where the uncertainties may reach 0.10, and for F11, j T〉 ) 0.5 mol · dm-3 when where the uncertainty is ≈0.20 at 〈C

646 Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 Table 9.

Tests of the ORR at T ) 298.15 K Using Equation 28a

compositionb

1012LHS/m5 · mol-1 · s-1

1 2 3 4 5

2.8657 2.8393 2.5801 2.3266 2.0239

6 7 8 9 10

1.4964 1.4939 1.4218 1.2923 1.1896

11 12 13 14 15

1.1541 1.1404 1.1021 1.0429 0.9290

16 17

1.0283 1.0445

18 19

1.0470 1.0458

20 21

1.1536 1.1240

22 23

1.2574 1.2474

1012RHS/m5 · mol-1 · s-1 j T〉 ) 0.5 〈C 2.9186 2.8031 2.6666 2.3462 2.0767 j T〉 ) 1.0 〈C 1.5871 1.5080 1.4367 1.3037 1.1822 j T〉 ) 1.5 〈C 1.1625 1.1639 1.1227 1.0352 0.9379 j T〉 ) 2.0 〈C 1.0635 1.0460 j T〉 ) 3.0 〈C 1.0352 1.0316 j T〉 ) 4.0 〈C 1.1262 1.1258 j T〉 ) 5.0 〈C 1.2517 1.2413

1012(LHS - RHS)/m5 · mol-1 · s-1

1012(uncertainty)/m5 · mol-1 · s-1

-3

mol · dm

-0.053 +0.036 -0.087 -0.020 -0.053

(0.220 (0.125 (0.161 (0.220 (0.321

-0.091 -0.014 -0.015 -0.011 +0.007

(0.110 (0.059 (0.073 (0.033 (0.039

-0.008 -0.023 -0.021 +0.008 -0.009

(0.072 (0.036 (0.028 (0.021 (0.017

-0.035 -0.002

(0.053 (0.025

+0.012 +0.014

(0.020 (0.012

+0.027 -0.002

(0.031 (0.032

+0.006 +0.006

(0.049 (0.052

mol · dm-3

mol · dm-3

mol · dm-3

mol · dm-3

mol · dm-3

mol · dm-3

R ) 8.3145 J · K-1 · mol-1 and T ) 298.15 K. These reported quantities from eq 28 are LHS ) {a11(D12)V + a12(D22)V}/RT and RHS ) {a21(D11)V + a22(D21)V}/RT. If the ORR for isothermal diffusion in a ternary system is obeyed, then LHS and RHS are equal within the uncertainties of determination of these quantities. b The numbers in this column denote the compositions as defined in Table 1. a

j T〉 ) 1.0 mol · dm-3 when z1 ) z1 ) 0.25 and is ≈0.11 at 〈C j T〉 ) 0.90. Also, 0.10 uncertainties are assumed for Fij at 〈C -3 (4.0 and 5.0) mol · dm . For the errors in the experimental-based (Dij)V, we first examined plots of these values as a function of z1 at constant j T〉 ) (0.5, 1.0, and 1.5) mol · dm-3; at higher concentra〈C tions, such plots are less informative because the diffusion data are limited to a narrow range of z1. These plots, given in Figures 1 and 2 of ref 19, are generally smooth, but the j T〉 ) 0.5 mol · dm-3 and z1 ) 0.75 seems value of (D12)V at 〈C -9 low by 0.03 · 10 m2 · s-1, and we thus assigned it this j T〉 uncertainty. Plots of (D21)V and (D11)V as a function of 〈C showed no significant scatter at z1 ) 0.90 but were significantly scattered at z1 ) 0.95. These plots (not shown) indicate that, at z1 ) 0.95, (D21)V is uncertain by ( j T〉 ) (0.5 to 2.0) mol · dm-3 and 0.004 · 10-9 m2 · s-1 from 〈C -9 2 -1 by about ( 0.002 · 10 m · s at higher concentrations. The corresponding (D11)V plot at z1 ) 0.95 indicated a scatter of about ( 0.006 · 10-9 m2 · s-1. The coefficients (D12)V and (D22)V at constant z1 show too much variation with the concentration to show scatter of this size. However, by a comparison of the statistical uncertainties in (Dij)V and the results from subset analysis,17-21 we estimate that δ(D12)V ≈ 4δ(D21)V and δ(D22)V ≈ (1/2)δ(D11)V. These uncertainty estimates were accepted. At all of the other compositions, we accepted the earlier “rule-of-thumb” that uncertainties in diffusion coefficients are roughly 4 times the statistical errors calculated by the propagation-of-errors method.11,36

Variation of the Diffusion Onsager Coefficients with the Solute Concentration and Equivalent Fraction As seen from the information listed in Tables 7 and 8, the values of (L11)R and (L22)R in both reference frames are all positive, whereas the cross-terms (L12)R and (L21)R are all negative. As expected, the values of (L11)V, (L22)V, (L11)0, and j T〉. In (L22)0 at any constant z1 all increase with increasing 〈C j T〉, the values of (L11)V and addition, at any constant value of 〈C (L11)0 are largest as z1 f 1, whereas those of (L22)V and (L22)0 are smallest. Correspondingly, the values of (L22)V and (L22)0 are largest as z1 f 0, whereas those of (L11)V and (L11)0 are smallest. That is, the values of (L11)V, (L22)V, (L11)0, and (L22)0 are largest when the composition fraction of the other component is smallest (i.e., as the composition approaches the binary solution of the major component) and decrease as the composition fraction of the other component increases (i.e., as the composition approaches the trace concentration in the binary solution of the other component). In general, at constant values j T〉, (L11)V is much greater than (L22)V and (L11)0 is much of 〈C greater than (L22)0, but, surprisingly, when z1 ) 0.25, (L11)V ≈ (L22)V and (L11)0 ≈ (L22)0. Values of the cross-term diffusion Onsager coefficients (L12)V, (L21)V, (L12)0, and (L21)0 at constant z1 all become increasingly j T〉. Curiously, these cross-term more negative with increasing 〈C j T〉 ) (0.5, 1.0, or quantities as a function of z1 at constant 〈C 1.5) mol · dm-3 all exhibit minima at z1 ≈ 0.5, i.e., when the chloride and sulfate concentrations are equal. However, many additional diffusion coefficient measurements at other values of z1 near 0.5 would be needed to determine whether this

Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 647

minimum always occurs at the same composition fraction or j T〉. whether it varies slightly with 〈C We now focus our attention on the solvent-fixed (Lij)0 because they are the most useful for the theory of electrolyte solutions. In addition to our experimental values at finite concentrations, we also include their values at infinite dilution in terms of expressions of the Nernst-Hartley type. The equations are in terms of equivalent conductances of the ions Λ0i and salt Λ0, and the equivalent fractions xi, where N ) C1 + 2C2, x1 ) C1/(C1 + 2C2), and x2 ) 2C2/(C1 + 2C2). At infinite dilution, these diffusion Onsager coefficient values are independent of the reference frame. The Nernst-Hartley expressions,7 rewritten for the common cation case, are

(L11)0 Λ01[Λ0 - x1Λ01] ) 3 x1N 10 (V z )2F2Λ0

(29a)

(L22)0 Λ02[Λ0 - x2Λ02] ) 3 x2N 10 (V z )2F2Λ0

(29b)

Λ01Λ02 (L12)0 )- 3 x1x2N 10 V1aν2az1az2aF2Λ0

(29c)

Λ0 ) x1Λ01 + x2Λ02 + Λ03

(30)

1a 1a

2a 2a

where Via is the stoichiometric coefficient of anion i; zia is the signed valence of anion i; F is Faraday’s constant; and Λ01, Λ02, and Λ03 are respectively the equivalent conductances of the Cland SO2anions and the Na+ common cation. Equations 4 29a-29c have most of the equivalent-fraction dependence removed, so they show less curvature than Lij or Lij/N. However, some curvature must remain when plotted against x1 because xi appears explicitly in the numerator of (Lii)0 and because x1 and x2 ) 1 - x1 appear implicitly in the Λ0 terms. These infinite dilution values are contained in Table 10 at the z1 molarity fraction corresponding to those of our experiments. The Λ0i values used are taken from Robinson and Stokes1 and whose values (after conversion from international to absolute ohms) are (76.312, 79.980, and 50.075) Ω-1 · cm2 · equiv-1 (where “equiv” denotes equivalent) respectively for Λ01, Λ02, and Λ03. There are several ways to plot our data, which include (Lij)0, (Lij)0/N, or (Lii)0/xiN and (Lij)0/xixjN (where i * j) plotted against j T〉. As noted above, the latter two quantities should have xi or 〈C less curvature. Equations 29a-29c suggest that plotting (Lii)0/xiN and (Lij)0/ j T〉 may minimize the curvature xixjN against x1 at various 〈C j T〉 concentrations. As suggested by eq 29c, even at the higher 〈C the plots of (Lij)0/xixjN should be “flatter” than the others, and it is found to be true. These plots against x1 are shown in Figures 2-4. There is a curious crossover with (L22)0/x2N plotted against j T〉 ) 0.5 mol · dm-3 curves. x1 for the infinite dilution and 〈C Furthermore, plots of (Lij)0 and (Lij)0/N against x1 all show j T〉. minima for all values of 〈C Table 10.

Figure 3. Values of {109 · RT(L12)0/x1x2N}/mol · equiv-1 · m2 · s-1 against j T〉/mol · dm-3, where N is the total equivalent x1 at constant total molarity 〈C concentration of the electrolyte mixture, x1 is the equivalent fraction of NaCl, and x2 is the equivalent fraction of Na2SO4 in the mixture [N ) C1 + 2C2; x1 ) C1/(C1 + 2C2); x2 ) 2C2/(C1 + 2C2)].

Plots of (Lii)0/xiN and (Lij)0/xixjN where i * j plotted against j T〉 at various values of x1 show much more curvature than 〈C in Figures 2 to 4, as can be seen in Figures 5-7. The plots j T〉 at any fixed value of x1 (equivalent of (L11)0/x1N against 〈C j T〉 and to plots at constant z1) all decrease with increasing 〈C with increasing x1. The (L11)0/x1N curves become steeper as the composition fraction of NaCl decreases. These curves at

Nernst-Hartley Values of RT(L11)0/x1N, RT(L22)0/x2N, and RT(L12)0/x1x2N at Infinite Dilution and T ) 298.15 Ka 9

a

Figure 2. Values of {109 · RT(L11)0/x1N}/mol · equiv-1 · m2 · s-1 against x1 j T〉/mol · dm-3, where N is the total equivalent at constant total molarity 〈C concentration of the electrolyte mixture and x1 is the equivalent fraction of NaCl in the mixture [N ) C1 + 2C2; x1 ) C1/(C1 + 2C2)].

10 RT(L11)0/x1N

109RT(L22)0/x2N

109RT(L12)0/x1x2N

mol · equiv-1 · m2 · s-1

mol · equiv-1 · m2 · s-1

mol · equiv-1 · m2 · s-1

z1

x1

0.925 05 1.033 50 1.304 40 1.630 90 1.861 10

0.500 44 0.471 51 0.399 22 0.312 08 0.250 65

-0.641 21 -0.639 60 -0.635 60 -0.630 78 -0.627 37

0.95 0.90 0.75 0.50 0.25

0.904 76 0.818 18 0.600 00 0.333 33 0.142 86

These are the limiting values of the diffusion Onsager coefficients when C1 and C2 vanish at molar ratios corresponding to z1.

648 Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009

Figure 4. Values of {109 · RT(L22)0/x2N}/mol · equiv-1 · m2 · s-1 against x1 j T〉/mol · dm-3, where N is the total equivalent at constant total molarity 〈C concentration of the electrolyte mixture, x1 is the equivalent fraction of NaCl, and x2 is the equivalent fraction of Na2SO4 in the mixture [N ) C1 + 2C2; x1 ) C1/(C1 + 2C2); x2 ) 2C2/(C1 + 2C2)]. Note the crossover of j T〉 ) 0.5 mol · dm-3 curves. the infinite dilution and 〈C

j T〉/ Figure 5. Values of {109 · RT(L11)0/x1N}/mol · equiv-1 · m2 · s-1 against 〈C mol · dm-3 at constant molarity fraction z1, where N is the total equivalent concentration of the electrolyte mixture and x1 is the equivalent fraction of NaCl in the mixture [N ) C1 + 2C2; x1 ) C1/(C1 + 2C2)]. The points at each fixed value of z1 were connected by straight lines.

x1 ) 0.90476 and x1 ) 0.81818 (z1 ) 0.95 and z1 ) 0.90) j T〉 ) 5 mol · dm-3; the curves at lower converge at about 〈C values of x1 also appear to be headed to this overlap point, but the measurements could not be made because of solubility limitations for Na2SO4 in the mixtures. j T〉 at The corresponding plots of (L12)0/x1x2N against 〈C j various values of x1 all increase with 〈CT〉 and x1, i.e., become j T〉 is especially steep less negative. The increase with 〈C j T〉 ) (0 and 0.5) mol · dm-3, where there are no between 〈C measurements to characterize the slopes. A few crossovers are apparent on these curves, but because of their small size, their significance is not clear. The negative values of this quantity were expected from the Nernst-Hartley infinite dilution values (Table 10). j T〉 at any fixed value of x1 The plots of (L22)0/x2N against 〈C (equivalent to plots at constant z1) all decrease with increasing

Figure 6. Values of {109 · RT(L12)0/x1x2N}/mol · equiv-1 · m2 · s-1 against j T〉/mol · dm-3 at constant molarity fraction z1, where N is the total 〈C equivalent concentration of the electrolyte mixture, x1 is the equivalent fraction of NaCl, and x2 is the equivalent fraction of Na2SO4 in the mixture [N ) C1 + 2C2; x1 ) C1/(C1 + 2C2); x2 ) 2C2/(C1 + 2C2)]. The points at each fixed value of z1 were connected by straight lines.

j T〉/ Figure 7. Values of {109 · RT(L22)0/x2N}/mol · equiv-1 · m2 · s-1 against 〈C mol · dm-3 at constant molarity fraction z1, where N is the total equivalent concentration of the electrolyte mixture and x2 is the equivalent fraction of Na2SO4 in the mixture [N ) C1 + 2C2; x1 ) C1/(C1 + 2C2); x2 ) 2C2/(C1 + 2C2)]. The points at each fixed value of z1 were connected by straight lines.

j T〉 but increase with increasing x1. Between 〈C j T〉 ) (0 and 〈C -3 0.5) mol · dm , where there are no measurements, the curves rise and then decrease (have a maximum) at x1 ) 0.14286 and j T〉 x1 ) 0.33333 and show a regular decrease with increasing 〈C when x1 ) 0.60, with a reversal in the slope at higher values of x1. The changes in the slope are not unexpected given that + Na2SO4 is partially associated as NaHSO4 (aq), whereas Na and Cl show little tendency to associate, but why the reversal in the initial slope occurs when x1 ) 0.6 (z1 ) 0.75) is not clear. j T〉 (not shown) Plots of (Lij)0 against x1 and (Lij)0 against 〈C may be suitable for interpolation, despite their curvature and j T〉 ) 0 for all minima. Note that all values of (Lij)0 are 0 at 〈C x1.

Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009 649

Test of the ORR Compared to Estimated Experimental Uncertainties A direct comparison of RT(L12)V with RT(L21)V and of RT(L12)0 with RT(L21)0 provides a measure of the validity of the ORR. To the three significant figures reported in Table 7, in three cases RT(L12)V ) RT(L21)V; in some other cases RT(L12)V is less negative than RT(L21)V, but in other cases it is more negative. Similarly, for the solvent-fixed reference frame in Table 8, in four cases RT(L12)0 ) RT(L21)0; in some other cases RT(L12)0 is less negative than RT(L21)0, but in other cases it is more negative. The values of the differences between RT(L12)R and RT(L21)R are generally small in both reference frames and show no obvious significant systematic variations with concentration. As seen in Tables 7 and 8, the values of RT(L12)V, RT(L21)V, RT(L12)0, and RT(L21)0 vary by a factor of 10 from -0.02 · 10-12 to -0.2 · 10-12 mol · m-1 · s-1 as the total concentration and composition fraction are changed. The differences between RT(L12)V and RT(L21)V and between RT(L12)0 and RT(L21)0 at the same composition are generally small, e0.003 · 10-12 mol · m-1 · s-1, in all but four cases for each reference frame. The other four cases with slightly larger deviations tend to occur at higher concentrations. As noted above, a more sensitive and quantitative test of the ORR (in which the quantities being compared only vary by a factor of 3) can be made by using the RHS and LHS of eq 28 for the volume-fixed reference frame, which we denote as LHS ) {a11(D12)V + a12(D22)V}/RT and RHS ) {a21(D11)V + a22(D21)V}/RT. Table 9 summarizes the values of the LHS and RHS of eq 28, their difference LHS - RHS (with sign), and the uncertainty of this difference calculated from the uncertainties of the input quantities using the propagation-of-errors method. At 8 compositions, the differences are small and positive, and at the other 15 compositions, they are small and negative. In 18 cases, the differences are well within the calculated uncertainties, thus indicating that the ORR is obeyed well within experimental error. In four other cases, the differences are slightly larger but still adequately within the assigned experimental error. At the j 1〉 ) 2.700 25 mol · dm-3 and 〈C j 2〉 only other compositions, 〈C -3 ) 0.300 04 mol · dm , the difference is 0.014 · 10-12 m5 · mol-1 · s-1, which only slightly exceeds the calculated uncertainty of 0.012 · 10-12 m5 · mol-1 · s-1. Given that the ORR for ternary-solution isothermal diffusion is obeyed within experimental uncertainty at 22 of 23 compositions, the small discrepancy at the other composition is likely just due to a slight underestimation of the experimental uncertainty of one or more of the diffusion coefficients.

Summary 17-21

We used previously reported precise diffusion and density measurements for the NaCl + Na2SO4 + H2O system at 298.15 K at 23 overall compositions spanning a wide composition range to test the ORR for isothermal diffusion. Realistic errors were assigned to the individual diffusion coefficients from cross-plotting and an earlier “rule-of-thumb” based on subset analysis for diffusion measurements at the same overall composition.11,36 Available isopiestic data24,33 for this system were reanalyzed with a hybrid thermodynamic model that was then used to calculate the chemical-potential concentration derivatives and estimate their uncertainties. These results were combined to calculate the diffusion Onsager coefficients on both the volume-fixed and solvent-fixed reference frames, (Lij)V and (Lij)0, respectively. The ORR for ternary-solution

diffusion was found to be obeyed within the assigned uncertainty limits at 22 of 23 compositions, and the very minor discrepancy at the other composition is probably due to a slight underestimation of the uncertainties for the diffusion coefficients. These results give very extensive and rigorous tests, and possibly the most accurate test, of the ORR for ternary-solution diffusion. Appendix

We briefly describe the relations between various descriptions of the Dij and Lij, their transformations, and equations needed for the ORR tests. These are obtained most transparently using the matrix methods described by De Groot and Mazur.37 We use the following notation taken from a comprehensive summary of previous work (containing the original references) showing how irreversible thermodynamics leads to the complete macroscopic description of liquid-state diffusion.4 For an arbitrary reference frame R

JR ) DRC

(A1)

J )L Y

(A2)

R

R

R

R

R

where J , C, and Y are the respective column vectors of the flows, negative concentration gradients, and driving forces of irreversible thermodynamics for reference frame R and where DR and LR are the respective square matrices of the diffusion coefficients and diffusion Onsager coefficients. Transformation of JR, LR, and YR between different reference frames requires that the “entropy production” of irreversible thermodynamics be invariant.4,37 The elements of the negative concentration gradient matrix C are

Ci ′ ) -∂Ci ⁄ ∂x

(A3)

R

The simplest Y is that for the solvent-fixed reference frame, Y0 ) X, whose elements are

Xi ) (∂Gi ⁄ ∂x)p,T

(A4)

The elements Xi of the X matrix can be written as

Xi )

∑ k

( )( ) ∂Ck ∂Gi ∂Ck ∂x

(A5)

or

X ) µC

(A6)

and where the elements of µ are

µik )

∂Gi ∂Ck

(A7)

SolWent-Fixed Reference Frame. For this reference frame,

J0 ) D0C ) L0X ) L0µC

(A8)

D0 ) L0µ

(A9)

L0 ) D0µ-1

(A10)

from which

and

The individual (Lij)0 are our desired diffusion Onsager coefficients and are given explicitly by eqs 23a to 23d. Unfortunately, for more than three components, the expressions for µ-1 contain increasingly complicated terms involving products of (n - 2)µij terms divided by the product of (n -

650 Journal of Chemical & Engineering Data, Vol. 54, No. 2, 2009

1)µij terms. Consequently, the calculations of the (Lij)0 terms are even more complicated. Volume-Fixed Reference Frame. For this reference frame,

JV ) DVC ) LVYV

(A11)

However, it has been shown4 that

YV ) A˜0VX ) A˜0VµC

(A12)

where the tilde represents the transpose of a matrix. Thus, the ˜ 0V are elements of A

A˜ij0V ) Rij ) Aji0V

(A13)

and where4

Aij0V ) δij +

∑ nk)1 µkiDkj0 ) ∑ nk)1 Dki0 µkj

This gives n equations for i ) j, whose results are identities. For i * j, there are (n2 - n)/2 nontrivial equations equivalent to the ORR but which have the great advantage that they are linear in µij for any number of components, unlike the direct comparison of the L0ij expressions themselves. This result for the ternary-solution solvent-fixed case was first given by Onsager3 and generalized by Hooyman and De Groot as described by De Groot and Mazur.37 As mentioned earlier, it is advantageous to work directly with the experimental DVij to minimize the estimated errors, i.e., within the volume-fixed reference frame. Thus, using eq A16 for LV, the ORR are

˜V LV ) DVµ-1(A˜0V)-1 ) L˜V ) (A0V)-1µ˜ -1D

jj CiV j0 C0V

(A14)

(A24)

By multiplying on the left by A0V and then by µ˜ and on the ˜ 0V and then µ, we get right by A

˜ VA ˜ 0Vµ µ˜ A0VDV ) D

Consequently,

(A25)

DV ) LVA˜0Vµ

(A15)

The resulting expressions are linear in µij for any number of components, unlike the Lij expressions themselves. The component form of eq A25 is

LV ) DVµ-1(A˜0V)-1

(A16)

∑ k ∑ l µkiAkl0V(Dlj)V ) ∑ k ∑ l (Dki)VAlk0Vµlj

and

V

V

The four (Lij) components of L are given explicitly in eqs 25a to 25d.10,14 We note that from the definition4 of ARS ij

(Aij0V)-1 ) AijV0 ) δij -

jj CiV 103

(A17)

which is ji of ref 14 (note the reversed subscript of ij). Again, unfortunately, for more than three components, the expressions for µ-1 contain increasingly complicated terms involving the products of µij. Consequently, the calculations of the (Lij)V terms are also more complicated. Other Matrix Results. It can be shown4 that

D0 ) A0VDV

µ)

( )

1 µmA0V C0M0

(A19)

m where, as noted earlier, the elements µm ij ) (∂Gi/∂mj)p,T of µ are related to the activity coefficient derivatives by eqs 14a and 14b. Transformed Test of the ORR. The matrix form of the ORR is

LR ) L˜R

(A20)

or ) The simplest example is the solvent-fixed case, where from eqs A10 and A20 LRji .

˜0 L0 ) D0µ-1 ) L˜0 ) µ˜ -1D

(A21)

If we multiply both sides of this equation on the left by µ˜ and on the right by µ, we get the equivalent equation

˜ 0µ µ˜ D0 ) D whose component form is

(A26)

There is one nontrivial ORR for a three-component system, three for a four-component one, and six for a five-component one. ˜ 0V, whose As noted earlier, the matrix of Rij equals the matrix A 0V components are Aji (note the reversed indices). Equation 28 follows from the above result. Acknowledgment We congratulate and dedicate this paper to Professor Robin Stokes on the occasion of his 90th birthday. His book (with R. A. Robinson) Electrolyte Solutions remains a constant reference source for our work, and in most areas of our research, Prof. Stokes has published significant pioneering research.

Literature Cited (A18)

whose component form was given in eq 20, which allows (Dij)0 to be calculated from the experimental (Dij)V. Furthermore, the relationship of µ and µm is

LRij

(A23)

(A22)

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Received for review September 29, 2008. Accepted November 15, 2008.

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